A295638 Take the sequence of nonnegative integers whose decimal digits are not in strictly increasing order. Partition the sequence into subsequences whose elements are consecutive integers. Then a(n) is the number of elements in the n-th partition.

2, 3, 4, 5, 6, 7, 8, 9, 33, 4, 5, 6, 7, 8, 9, 44, 5, 6, 7, 8, 9, 55, 6, 7, 8, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 444, 5, 6, 7, 8, 9, 55, 6, 7, 8, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 555, 6, 7, 8, 9, 66, 7, 8, 9, 77, 8, 9, 88, 9, 99, 666, 7, 8

Offset: 1

Author

Gunnar Lee Johnson, Nov 24 2017

Keywords

• nonn
• fini
• less
• base

Comments

Only defined as an integer for a(1) through a(255), as a(256) references the infinite partition (123456790, 123456791, ..., 999999999, 1000000000, 1000000001, ...). No integer greater than 123456789 has a strictly increasing sequence of digits (itself being the only case for 9 digits, and by the pigeonhole principle, a >9-digit number must have a digit repeated and is thus not strictly increasing).

Examples

			For a(1)=2 through a(8)=9, these correspond to the consecutive subsequences (10, 11), (20, 21, 22), ..., (80, 81, 82, ..., 88). The jumps at e.g. a(9)=33 or a(37)=444 correspond to (90, 91, ..., 122) and (790, 791, ..., 1233), where 89 and 123, and 789 and 1234, are the values partitioning the subsequences.
		

Links

• Gunnar Lee Johnson, Table of n, a(n) for n = 1..255

Crossrefs

The nonnegative integers minus A009993 is the sequence that is partitioned.

Programs

PARI

is(n) = my(d=digits(n)); d != vecsort(d,,8);
lista(nn) = {my(w = select(n->is(n), vector(nn, k, k))); my(dw = vector(#w-1, k, w[k+1] - w[k])); my(k = 1); for (n=1, #dw, if (dw[n] == 1, k++, print1(k, ", "); k = 1););} \\ Michel Marcus, Jan 08 2018

Python

def a(n):
    (x,i,count,switch) = (0,0,1,True)
    while True:
        if switch == (list(sorted(set(str(i)))) == list(str(i))):
            count += 1
        else:
            if not switch: x += 1
            if x == n: return count
            (count, switch) = (1, not switch)
        i += 1